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Frequentist properties of Bayesian inequality tests

David M. Kaplan, Longhao Zhuo

Abstract

Bayesian and frequentist criteria fundamentally differ, but often posterior and sampling distributions agree asymptotically (e.g., Gaussian with same covariance). For the corresponding single-draw experiment, we characterize the frequentist size of a certain Bayesian hypothesis test of (possibly nonlinear) inequalities. If the null hypothesis is that the (possibly infinite-dimensional) parameter lies in a certain half-space, then the Bayesian test's size is $α$; if the null hypothesis is a subset of a half-space, then size is above $α$; and in other cases, size may be above, below, or equal to $α$. Rejection probabilities at certain points in the parameter space are also characterized. Two examples illustrate our results: translog cost function curvature and ordinal distribution relationships.

Frequentist properties of Bayesian inequality tests

Abstract

Bayesian and frequentist criteria fundamentally differ, but often posterior and sampling distributions agree asymptotically (e.g., Gaussian with same covariance). For the corresponding single-draw experiment, we characterize the frequentist size of a certain Bayesian hypothesis test of (possibly nonlinear) inequalities. If the null hypothesis is that the (possibly infinite-dimensional) parameter lies in a certain half-space, then the Bayesian test's size is ; if the null hypothesis is a subset of a half-space, then size is above ; and in other cases, size may be above, below, or equal to . Rejection probabilities at certain points in the parameter space are also characterized. Two examples illustrate our results: translog cost function curvature and ordinal distribution relationships.

Paper Structure

This paper contains 37 sections, 4 theorems, 72 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Contributions
  4. Paper structure and notation
  5. Literature
  6. Bayesian--frequentist comparisons
  7. More closely related papers
  8. Setup and assumptions
  9. The Bayesian test and decision-theoretic context
  10. Assumptions
  11. Posterior and sampling distributions
  12. Shape of null hypothesis region
  13. Least favorable prior approach
  14. Results and discussion
  15. Intuition for results
  16. ...and 22 more sections

Key Result

Theorem 1

Let a:multia:H0 hold. Consider the Bayesian test of meth:Bayes.

Figures (1)

  • Figure 1: Illustrations of \ref{['thm:1']} for $\boldsymbol{\mathbf{X}},\boldsymbol{\mathbf{\theta}}\in{\mathbb R}^2$.

Theorems & Definitions (8)

  • Theorem 1
  • proof : Proof of \ref{['thm:1']}
  • Proposition 2
  • Proposition 3
  • Corollary 4
  • proof : Proof of \ref{['prop:SD1-PrH0']}
  • proof : Proof of \ref{['prop:SD1-size']}
  • proof : Proof of \ref{['cor:nonSD1-RP']}